Every time you do a pinch flex on a hexaflexagon made from equilateral triangles, it takes you from one hexagon position to another.
The pinch flex rotates the leaves, placing a different set of vertices in the center.
With an equilateral triangle, this always gives you a hexagon.
But when the triangles aren't equilateral, the pinch flex can give you two or three different arrangements.

A bronze hexaflexagon provides good examples of the three main possible arrangements.
Typically you first fold it into a flat triangle, with six 60 degree angles meeting in the middle.
But pinch flexing in one place will give you a cup, with six 30 degree angles meeting together, which can't be folded out flat.
And pinch flexing in a different place will give you a flower shape, with six 90 degree angles meeting in the center, which won't lie flat either.

It's interesting to note, however, that both the cup and flower positions can still allow pinch flexes.
Some may not work from a cup and the flower may require an interesting shift between valley and mountain folds first.

Typically the pinch flex maintains the symmetry of sectors, i.e. adjacent pairs of pats (stacked triangles).
But the isosceles enneaflexagon demonstrates a case where a sector is actually three pats in a row.

The heptaflexagon is an example of where it's useful to perform the pinch flex on a just subset of the flexagon.
In this case, it means that you have an extra little tent after flexing.
But the same thing can be done on higher order flexagons as well, leaving larger portions of the flexagon unchanged.